정지된 유체속에 분산되어 있는 캡슐의 전기수력학적 변형에 관한

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하종욱, 양승만

한국과학기술원  화학공학과

Small deformation theory on the Electrohydrodynamic Deformation of a capsule neutrally suspended in a quiescent fluid

Jong-Wook Ha, Seung-Man Yang

Department of Chemical Engineering

Korea Advanced Institute of Science and Technology

INTRODUCTION

Barther-Biesel [1] has proposed the use of the term 'capsule' to describe a incompressible Newtonian drop which is enclosed by elastic or viscoelastic membrane. Such a particle provides a mathematical model to describe the flow-induced deformation of red blood cells, emulsions stabilized by layers of surface active agents and of microcapsules produced by interfacial cross-linking polymerization in drug delivery system. Depending on the assumptions made regarding the physical properties of the internal fluid or of the interface, a capsule may serve as a model for a number of particles. For example, if both the internal and suspending fluids are immiscible liquids and the interface is simply the liquid-liquid interface, the a capsule is just an ordinary liquid droplet and the study of its motion and deformation is relevant for a good understanding of emulsion rheology. The interface may also be a thin deformable solid, in which case the particle is an artificial capsule, such as those commonly used in some technological applications. The basic idea of the present study is the fact that the electric-field-induced flow, which is analogous to axisymmetric (uniaxial or biaxial) straining flow in qualitative, can be generated around the particle. The interest here is focused on the influence of the interface on the overall behavior of a capsule in a uniform electric field, and consequently the discussion is restricted to those capsules which are initially spherical and filled with a Newtonian liquid.

GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

We assume that there is no external mean flow field, and therefore, in the absence of electric field, the fluids are quiescent. The mechanical deforming forces are due to the electric stresses and viscous stresses exerted by the electric-field-induced flow. The interface is considered to be infinitely thin, so that it has no bending resistance, and to be characterized by different macroscopic constitutive equations. The fluid mechanical problems of the flows of the internal and external liquids are solved under the condition of kinetic and dynamic equilibrium of the membrane. This is done by means of a regular perturbation analysis developed in the case where the deformation is small.

Consider a capsule of undeformed radius a, filled with an incompressible liquid of viscosity [eqn: lambda mu], electrical resistivity [eqn: R chi], and permittivity [eqn: S epsilon] and surrounded by a two-dimensional membrane with a surface shear elastic modulus [eqn: E^s], and a surface viscosity [eqn: mu ^s]. This capsule is suspended in another Newtonian incompressible liquid of viscosity [eqn: mu], resistivity [eqn: chi], and permittivity [eqn: epsilon], which is subjected to a uniform electric field strength V. In order to non-dimensionalize, lengths are scaled by a, electric and viscous stresses by [eqn: epsilon V^2] and time by [eqn: mu / epsilon V^2]. Assuming the capsule to be small, the Reynolds number based on its dimensions is much smaller than unity, so that inertial effects may be neglected. The dynamics of electric-field-induced flow is described by the Stokes equations. The non-dimensional governing equations with respect to a set of axes moving with the center of mass of the cell are:

[eqn: TRIANGLED CDOT {BOLD v} = 0], [eqn: TRIANGLED CDOT {BOLD sigma } = 0],                                               (1)

in which [eqn: {BOLD sigma } = -p{BOLD I} +( TRIANGLED {BOLD v}+ TRIANGLED {BOLD v} ^t )+{BOLD sigma }^E] for outside the capsule. The boundary conditions are

[eqn: {BOLD v}~ ->~ 0] as [eqn: LEFT | {BOLD x} RIGHT |~ -> ~ INF],                                              (2)

[eqn: LEFT [ {BOLD v} RIGHT ] _{ PARTIAL s}~=~0],                                                        (3)

[eqn: LEFT [ {BOLD sigma } CDOT {BOLD n} RIGHT ] _{ PARTIAL s} + {BOLD f}~=~0],                                                (4)

where, f is the force exerted by the membrane on the fluids and depends on the wall mechanics, which has to be specified.

SPECIFICATION OF CAPSULE DISTORTION

By analogy with the problem of determining the deformation of a surface-tension drop in an electric field, we seek a deformation field which involves only second harmonics at leading order and consider

[eqn: {BOLD x}~ = {BOLD X}+ varpi {BOLD K} CDOT {BOLD X}+varpi {BOLD XX} CDOT ( {BOLD J}-{BOLD K}) CDOT {BOLD X}+O(varpi^2 )],                 (5)

in which J and K are function of time only. The instantaneous external shape of the capsule is then

[eqn: r~ ==~ LEFT | {BOLD x} RIGHT | =1+varpi {BOLD X} CDOT {BOLD J} CDOT {BOLD X} + O(varpi^2 )= 1+varpi {BOLD x} CDOT {BOLD J} CDOT {BOLD x}/r^2 +O(varpi^2 )],     (6)

indicating that at leading order in [eqn: varpi {BOLD K}] represents motion of fibers within the surface, whereas J involves overall deformation of the surface. Accordingly, the overall distortion of the profile of the particle is determined by J, whereas K measures in-plane deformations of the membrane elements. We suppose further that J, and K are symmetric and traceless second-rank tensors, so that they involve only second-harmonic distortions. In that case of J, symmetry may be imposed without loss of generality, and tracelessness follows from conservation of the drop volume. For K we cheak a posteriori that the equations for its time evolution preserved these requirement.

In the case of a viscoelastic membrane, because of the linearizing procedure, to first order in [eqn: varpi], the viscous behavior simply adds up to the elastic one. Consequently, the time evolution of the deformation is

[eqn: beta {DOT {BOLD J}}=(2 lambda +3)(a_0 + a_1 )(1+ beta ){BOLD E} + {BOLD L} + {5 OVER 8}{BOLD M}],                          (7)

[eqn: beta {DOT {BOLD K}}={1 OVER 2} (2 lambda +3)(a_0 + a_1 ) (1+ beta) {BOLD E} + {1 OVER 2} {BOLD L}+ {1 OVER 4} {BOLD M}],                    (8)

where

[eqn: a_0 = {{9(1-R^2 )(3lambda+2)} OVER {(2R+1)^2 (16+19lambda) (2lambda+3)}}-{{9R(RS-1)} OVER {(2R+1)^2 (2lambda+3)}}] ,

and, [eqn: a_1 = {{9(1-R)^2} OVER {(2R+1)^2 (16+19lambda)}}] .

E is the rate-of-strain tensor caused by electric-field-induced flow. In addition, L and M are both symmetric and traceless and so involve only second harmonics. They can be specified by the choice of the load vector f. The new parameter [eqn: beta (= mu ^s varpi V^2 / E^s )] is the ratio of the characteristic relaxation time of the membrane [eqn: mu ^s / E^s] to the time [eqn: 1/varpi V^2]. In fact, it is identical with the Deborah number, commonly used for viscoelastic fluids, which measures the relative magnitudes of the flow time and of the liquid time.

Since the electric-field-induced flow is purely axisymmetric, the flow has a zero vorticity. At this order, Jaumann derivatives appeared in above equations can be replaced by time derivative [eqn: PARTIAL / PARTIAL t], but at the higher order correction the contribution from vorticity is apparent.

SOLUTIONS

1. Pure surface tension membrane

One of particular cases which is included in the model is the surface tension droplet. Then the membrane surface energy is proportional to its area and [eqn: {BOLD L} = 0], [eqn: {BOLD M} = -4 gamma {BOLD J}]. At equilibrium,

[eqn: { {PARTIAL {BOLD J}} OVER {PARTIAL t}}=0] so that [eqn: {BOLD J} = {{(2 lambda +3)(19 lambda +16)} OVER {40(1+ lambda )}}(a_0 + a_1 ){BOLD E}]                 (9)

but plainly [eqn: PARTIAL {BOLD K} / PARTIAL t~ !=~ 0]: there is still continuous shearing of the fibers within the membrane.

2. Three-dimensional isotropic incompressible membrane

Here, it is assumed that the elasticity of the membrane corresponds to that of an infinitely thin sheet of a three-dimensional isotropic, incompressible material. This leads to the following values of the L and M:

[eqn: {BOLD L} = 4 {BOLD J} - {22 OVER 3} {BOLD K}]

[eqn: {BOLD M} = -8 {BOLD J} + {40 OVER 3} {BOLD K}]

Then, the general solution is straightforward:

[eqn: J_{11} =-{5 OVER 6} (2 lambda +3)(a_0 + a_1 ) (1+ beta) +z_1 e^{-t/beta} +z_2 e^{-t/3beta}]                  (10)

[eqn: K_{11} = -{1 OVER 2} (2lambda +3) (a_0 + a_1 )(1+beta)+ z_2 e^{-t/beta}]                            (11)

The dynamic response of the particle has two relaxation times [eqn: beta] and [eqn: 3beta]. This is due to the fact that two independent modes of deformation are allowed for the surface: pure shear and area dilation.

3. The RBC-type elasticity

It is assumed now that the elastic properties of the material are similar to those of a red blood cell. This means that the membrane strongly resists area changes, but has a relatively low shear modulus. Following Barthes-Biesel and Rallison [2], we introduce a second-order tensor F, which is defined by

[eqn: {BOLD J} = {3 OVER 2} {BOLD K} + {BOLD F}]

Then eliminating [eqn: alpha _2 {BOLD F}] terms from the system, we obtaine a single differential equation for J:

[eqn: beta {{ PARTIAL {BOLD J}} OVER {PARTIAL t}}={3 Over 4} (2lambda +3)(a_0 +a_1 ) (1+beta) {BOLD E} - {1 OVER 2} {Bold J}]                            (12)

Thus the capsule, being restricted in the deformations it can achieve, has only one relaxation time, equal to [eqn: 2 beta]. The solution is

[eqn: J_{11}=-{1 Over 2} (2lambda+3) (a_0 + a_1 ) (1+beta) +z_1 e^{-t/2beta}]                            (13)

CONCLUSION

Purely electric-field-induced deformation of liquid capsule with a viscoelastic membrane is considered in the small deformation region. The flow induced by electric field is analyzed on the basis of leaky dielectric theory. Through the linearization of the governing equations and boundary conditions, the first correction to the shape deviated from initial spherical shape can ne calculated analytically. The behavior of viscoelastic membrane is described by means of a Kelvin-Voigt model. Steady shapes of the capsule are obtained for incompressible isotropic membrane and for RBC-type (area-preserving) membrane as well as for surface tension droplet. Unlike the deformation of capsules in a shear flow, the steady-shape of the capsules in an electric field depends on the viscosity ratio. Due to the vorticity-free nature of electric-field-induced flow, the axis of symmetry of the capsule is parallel to the direction of electric field.

REFERENCES

[1] Barthes-Biesel, D., J. Fluid Mech. 100, 831 (1980)

[2] Barthes-Biesel, D. and Rallison, J. M., J. Fluid Mech. 113, 251 (1981)